{"id":5339,"date":"2010-11-13T03:08:21","date_gmt":"2010-11-13T03:08:21","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5339"},"modified":"2022-01-12T01:37:57","modified_gmt":"2022-01-12T01:37:57","slug":"rascunho-22","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5339","title":{"rendered":"Averigue se o tri\u00e2ngulo [ABC] \u00e9 tri\u00e2ngulo ret\u00e2ngulo"},"content":{"rendered":"<p><ul id='GTTabs_ul_5339' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5339' class='GTTabs_curr'><a  id=\"5339_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5339' ><a  id=\"5339_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_5339'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Averigue se o tri\u00e2ngulo [ABC] \u00e9 tri\u00e2ngulo ret\u00e2ngulo e is\u00f3sceles, sendo:<\/p>\n<ol>\n<li>$A(1,1,\\sqrt{2})$, $B(\\sqrt{2},-\\sqrt{2},0)$ e C o sim\u00e9trico de A em rela\u00e7\u00e3o a O, origem do referencial;<\/li>\n<li>$A(2,1,-3)$, $B(-1,3,4)$\u00a0e $C(-3,0,2)$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5339' onClick='GTTabs_show(1,5339)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5339'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, $C(-1,-1,-\\sqrt{2})$.\n<p>Como<\/p>\n<p>$\\left\\| \\overrightarrow{AB} \\right\\|=\\sqrt{{{(\\sqrt{2}-1)}^{2}}+{{(-\\sqrt{2}-1)}^{2}}+{{(-\\sqrt{2})}^{2}}}=\\sqrt{{{(-1-\\sqrt{2})}^{2}}+{{(-1+\\sqrt{2})}^{2}}+{{(-\\sqrt{2})}^{2}}}=\\left\\| \\overrightarrow{BC} \\right\\|$,<br \/>\nent\u00e3o $\\overline{AB}=\\overline{BC}$.<\/p>\n<p>Como<br \/>\n$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{AB}.\\overrightarrow{BC} &amp; = &amp; (-1+\\sqrt{2},-1-\\sqrt{2},-\\sqrt{2}).(-1-\\sqrt{2},-1+\\sqrt{2},-\\sqrt{2})\u00a0 \\\\<br \/>\n{} &amp; = &amp; (-1+\\sqrt{2})(-1-\\sqrt{2})+(-1-\\sqrt{2})(-1+\\sqrt{2})+{{(-\\sqrt{2})}^{2}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; {{(-1)}^{2}}-{{(\\sqrt{2})}^{2}}+{{(-1)}^{2}}-{{(\\sqrt{2})}^{2}}+2\u00a0 \\\\<br \/>\n{} &amp; = &amp; 1-2+1-2+2\u00a0 \\\\<br \/>\n{} &amp; = &amp; 0\u00a0 \\\\<br \/>\n\\end{array}$<\/p>\n<p>ent\u00e3o os vetores s\u00e3o perpendiculares e, portanto, $A\\hat{B}C=90{}^\\text{o}$.<\/p>\n<p>Logo, o tri\u00e2ngulo [ABC] \u00e9 ret\u00e2ngulo em B e is\u00f3sceles.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Ora,\n<p>$\\overrightarrow{AB}=(-3,2,7)$ e $\\left\\| \\overrightarrow{AB} \\right\\|=\\sqrt{{{(-3)}^{2}}+{{2}^{2}}+{{7}^{2}}}=\\sqrt{62}$<\/p>\n<p>$\\overrightarrow{BC}=(-2,-3,-2)$ e $\\left\\| \\overrightarrow{BC} \\right\\|=\\sqrt{{{(-2)}^{2}}+{{(-3)}^{2}}+{{(-2)}^{2}}}=\\sqrt{17}$<\/p>\n<p>$\\overrightarrow{AC}=(-5,-1,5)$ e $\\left\\| \\overrightarrow{BC} \\right\\|=\\sqrt{{{(-5)}^{2}}+{{(-1)}^{2}}+{{5}^{2}}}=\\sqrt{51}$<\/p>\n<p>Logo, o tri\u00e2ngulo [ABC] n\u00e3o \u00e9 is\u00f3sceles.<\/p>\n<p>Por outro lado:<\/p>\n<p>$\\overrightarrow{AB}.\\overrightarrow{BC}=(-3,2,7).(-2,-3,-2)=6-6-14=-14\\ne 0$<\/p>\n<p>$\\overrightarrow{AB}.\\overrightarrow{AC}=(-3,2,7).(-5,-1,5)=15-2+35=48\\ne 0$<\/p>\n<p>$\\overrightarrow{BC}.\\overrightarrow{AC}=(-2,-3,-2).(-5,-1,5)=10+3-10=3\\ne 0$<\/p>\n<p>Logo, o tri\u00e2ngulo [ABC] n\u00e3o \u00e9 ret\u00e2ngulo, pois $\\begin{array}{*{35}{l}}<br \/>\nA\\hat{B}C\\ne 90{}^\\text{o} &amp; \\wedge\u00a0 &amp; B\\hat{A}C\\ne 90{}^\\text{o} &amp; \\wedge\u00a0 &amp; A\\hat{C}B\\ne 90{}^\\text{o}\u00a0 \\\\<br \/>\n\\end{array}$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5339' onClick='GTTabs_show(0,5339)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Averigue se o tri\u00e2ngulo [ABC] \u00e9 tri\u00e2ngulo ret\u00e2ngulo e is\u00f3sceles, sendo: $A(1,1,\\sqrt{2})$, $B(\\sqrt{2},-\\sqrt{2},0)$ e C o sim\u00e9trico de A em rela\u00e7\u00e3o a O, origem do referencial; $A(2,1,-3)$, $B(-1,3,4)$\u00a0e $C(-3,0,2)$. Resolu\u00e7\u00e3o &gt;&gt;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19175,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,111],"series":[],"class_list":["post-5339","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-geometria-analitica","tag-11-o-ano","tag-geometria","tag-vectores"],"views":1738,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat66.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5339","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5339"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5339\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5339"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5339"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5339"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5339"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}